Generalised massive gravity one-loop partition function and AdS/(L)CFT

TL;DR

The paper computes the Euclidean 1-loop partition function for Generalised Massive Gravity (GMG) on thermal AdS$_3$ and demonstrates a precise match with AdS$_3$/CFT expectations at several special points, including cases that map to logarithmic CFTs (LCFTs) and Conformal Chern–Simons gravity (CSG). By decomposing fluctuations with an auxiliary-field formulation and York decomposition, the authors derive a compact factorisation $Z_{ m GMG}=Z_E\,Z_{m_1}\,Z_{m_2}$, and they explicitly evaluate the determinants to expose the CFT content: the Virasoro vacuum, log partners, and massive primaries, along with the conformal ghost in the CSG limit. The results provide strong evidence that GMG at critical loci is dual to LCFTs with well-defined multiplicities $N_{h,\bar h}$ and that CSG realizes a reducible Verma module via a semi-classical null vector at level two. The work further suggests a generalisation to extended higher-derivative gravities consistent with a holographic $c$-theorem, predicting that their 1-loop determinants agree with GMG, thereby broadening the holographic dictionary for 3D gravity theories.

Abstract

The graviton 1-loop partition function is calculated for Euclidean generalised massive gravity (GMG) using AdS heat kernel techniques. We find that the results fit perfectly into the AdS/(L)CFT picture. Conformal Chern-Simons gravity, a singular limit of GMG, leads to an additional contribution in the 1-loop determinant from the conformal ghost. We show that this contribution has a nice interpretation on the conformal field theory side in terms of a semi-classical null vector at level two descending from a primary with conformal weights (3/2,-1/2).

Figures (1)

• Figure 1: Real (a) and imaginary (b) parts of $\ln Z_{\rm CSG}$. The shading goes from darker (lower values) to brighter (higher values). The plots are cut off at large absolute values, so the white regions along the unit circle represent poles of either sign.